Divisibility of Lee's class and its relation with Rasmussen's invariant

TL;DR

This work generalizes Lee-type link homology by introducing the c-divisibility k_c of the Lee class within the family H_c(-;R) parameterized by c = \\sqrt{h^2 + 4t}. It defines the invariant \\overline{s}_c(L) = 2k_c(D) + w(D) - r(D) + 1 and proves its well-definedness as a link invariant, along with cobordism functoriality and concordance-type bounds. The authors provide substantial evidence that \\overline{s}_c agrees with Rasmussen's knot invariant s, including a direct knot-case equivalence for (R,c) = (F[h],h) where s(K;F) = \\overline{s}_h(K;F[h]); they also develop a unifying parameter-reduction framework linking various theories (Khovanov, Lee, Bar-Natan) and show the potential to transfer results like the Milnor conjecture through this invariant. The paper closes by sketching connections to transverse invariants, Jones-type conjectures, and a set of open questions about the universality of s across fields and coefficients.

Abstract

Lee homology (a variant of Khovanov homology) over $\mathbb{Q}$ possesses the "canonical generators" as its basis. The generators (Lee's classes) $[α(D, o)]$ are constructed combinatorially from an oriented link diagram $D$, one for each alternative orientation $o$ on $D$. Let $R$ be an integral domain. There exists a family of link homology theory $\{ H_c(-; R) \}_{c \in R}$, where Khovanov's theory corresponds to $c = 0$ and Lee's theory corresponds to $c = 2$. For each $c \in R \setminus 0$, Lee's classes $[α(D, o)]$ can be defined as elements in $H_c(D; R)$, but when $c$ is not invertible then they do not form a basis; in fact they are divisible by $c$-powers. We define the $c$-divisibility $k_c(D)$ of $[α(D, o)]$ with $o$ the given orientation of $D$. For any link $L$ and its diagram $D$, we prove that $\bar{s}_c(L) := 2k_c(D) + w(D) - r(D) + 1$ is a link invariant, where $w$ is the writhe, and $r$ is the number of Seifert circles. We pose the question whether $\bar{s}_c$ coincides with Rasmussen's $s$-invariant. There are several evidences that support the affirmative answer. For instance, $\bar{s}_c$ is a link concordance invariant, and the Milnor conjecture can be reproved using $\bar{s}_c$. Also for the special case $(R, c) = (\mathbb{Q}[h], h)$, our $\bar{s}_c$ actually coincides with $s$ as knot invariants.

Figures (25)

• Figure 1: The $(h, t)$-parameter space.
• Figure 2: 0-, 1-resolution of a crossing.
• Figure 3: $\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu^\cdot(D)$ for the left hand trefoil.
• Figure 4: Orientation preserving resolution.
• Figure 5: Coloring the Seifert circles by $\mathbf{a}$, $\mathbf{b}$.

Theorems & Definitions (107)

• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 2.1: khovanov2004
• Proposition 2.2
• Proposition 2.3
• Lemma 2.6